Time series analysis 1

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What is the basic idea of time series data?

Time series data are collected on the same observational unit at multiple time periods

What are time series models often not based on?

Theoretical economic models

What are time series models usually indexed with?

t, time

What is the formal definition of a white noise process?

$E(y_t) = \mu$, $var(y_t) = \sigma^2$, $cov(y_t, y_{t-j}) = 0$ for all $j = 1, 2, 3, \ldots$

What is the null hypothesis in the Ljung-Box Q-test for autocorrelation?

No autocorrelation at the $j$ first lags

What does the autocorrelation function (ACF) indicate about the stationarity of a time series?

Autocorrelations of non-stationary series decay very slowly

What is the best forecast for $y_{t+1}$ in a white noise process with constant mean?

$\mu$

What is the conventional definition used for time series data when calculating the autocorrelation function?

The summation is over $t=j+1$ to $T$. The divisor is $T$, not $T – j$

What does the ACF computation assume about the process?

Stationarity of the process

What information does the ACF provide for nonstationary series?

Autocorrelations decay very slowly

What is the implication of positive autocorrelation in a time series?

Momentum effect

Which concept is commonly referred to in the literature when discussing stationarity in time series analysis?

Covariance stationarity

What is a key reason for using natural logs in time series analysis?

Closer approximation to normal distribution

What is the primary focus of macroeconometrics?

Forecasting models and their limitations

What is the purpose of using time series data in economic analysis?

Estimating dynamic causal effects

In what form have the Real vs. nominal housing price indices for the Helsinki Metro Area been observed?

Natural log form

What is a technical issue raised by time series data in economic analysis?

Calculation of standard errors for serially correlated errors

What is the focus of macroeconomists when using time series data?

Economic fundamentals like GDP, consumption, household indebtedness, inflation, and interest rates

What is a persistent, long-term movement or tendency in the data called?

Deterministic trend

Which type of time series becomes stationary after removing a deterministic trend from it?

Trend stationary

What is a stochastic trend with serially uncorrelated disturbances called?

Random walk

What is the correlation of a series with its own lagged values called?

Autocorrelation

What type of time series are called I(0)?

Stationary

What type of time series are called I(1)?

Difference stationary

What type of time series are called I(2)?

Difference stationary

What is a nonrandom function of time called?

Deterministic trend

What is a random and varies over time called?

Stochastic trend

What should be used for regression analysis if a time series has a random walk trend?

$ΔYt$

What are estimates of the population autocorrelations computed over observations in the time series called?

Sample autocorrelations

What is the correlation of a series with its own lagged values called?

Autocorrelation

In the context of time series modelling, what is the primary assumption underlying the AR(p) model?

The previous and current values of the time series contain predictive power for future values.

What type of time series memory is associated with the long-term effect of a shock gradually disappearing over time?

Long term (AR)

What is the distinguishing feature of short-term memory in time series processes?

The market immediately corrects itself after a shock.

What is the Box-Jenkins estimation approach primarily focused on?

Estimating and analyzing ARIMA models for time series data.

What does the Yule-Walker equations solve for in an AR model?

The autocorrelations and AR coefficients

In an AR model, what is the implication of a negative coefficient?

The series bounces from one side of its mean to the other

What condition makes an AR model stationary?

The sum of autoregressive coefficients does not equal 1

What does the reaction to a shock in an AR model depend on?

The coefficient

What do non-stationary AR processes lead to?

Explosive behavior in the series

What do the partial autocorrelation function measure in an AR model?

The correlation between an observation k periods ago and the current observation

What is the primary focus of an AR(p) model?

Predicting/explaining yt using its own previous values

What influences the behavior of the series in an AR model?

The mean, the swings around the mean, and the autocorrelation in the series

What does a greater coefficient in an AR model imply?

The longer-lasting effect of a shock

What can capture seasonal variation in yt in an AR model?

Seasonal dummies

What is the predicted value of yt calculated using in a simple AR(1) model?

The parameter estimates

What can be used to solve for the autocorrelations and AR coefficients in an AR model?

Yule-Walker equations

What is commonly used for model selection in econometric modeling?

Information criteria such as Akaike and Schwartz

What is the implication of non-normality of the error term in econometric modeling?

It leads to unreliable confidence intervals and p-values

What is the main focus of diagnostic checks in econometric modeling?

To detect whether the model reflects the DGP well and if the error term is white noise

What is the primary consideration for selecting the best model in econometric modeling?

Model fit

What is the recommended approach to controlling non-normality of the error term in econometric modeling?

Using point dummy variables

What is the purpose of comparing models using the same sample period and fixed T in econometric modeling?

To ensure consistency in estimation

What is the primary focus of maximum likelihood estimation in econometric modeling?

To estimate the parameters that maximize the likelihood of the observed data

What is the reason for not recommending overdifferencing in econometric modeling?

Valuable information is lost

What is the primary consideration for selecting parsimonious models in econometric modeling?

They typically produce more accurate forecasts

What is the focus of formal testing such as the Jarque-Bera test in econometric modeling?

To check for residual normality

Why is testing for autocorrelation particularly important in econometric modeling?

Evidence of residual autocorrelation implies that the model does not cater for some systematic process in the time series

What is the primary purpose of visual inspection in econometric modeling?

To detect outlier observations

What is the distinguishing feature of an autoregressive model (AR(p))?

It assumes that previous and current values of time series contain predictive power w.r.t. future values

Which type of time series memory is associated with the long-term effect of a shock gradually disappearing over time?

Long term (AR)

What is the primary focus of the Box-Jenkins estimation approach mentioned in the handout?

Model estimation and diagnostic checking

What type of time series modelling is based on the assumption that the market immediately corrects itself after a shock?

Moving average processes (MA models)

Which method is commonly used for estimation in the identification stage of econometric modeling?

Maximum Likelihood estimation

What is the purpose of diagnostic checks in econometric modeling?

To detect model estimation errors

Which information criteria are commonly used for model selection in econometric modeling?

Akaike Information Criterion (AIC) and Schwarz Criterion (SC)

What do non-normality of the error term lead to in econometric modeling?

Unreliable confidence intervals and p-values

What do diagnostic checks in econometric modeling aim to detect?

White noise in the error term

What is the implication of residual autocorrelation in a time series model?

The model caters for systematic processes

What is the primary focus of comparing models using the same sample period and fixed T in econometric modeling?

Ensuring fair model comparison

What is the primary focus of formal testing such as the Jarque-Bera test in econometric modeling?

Testing for residual normality

What is the purpose of using point dummy variables or the Student's t-distribution in econometric modeling?

To control non-normality of the error term

What is the conventional approach for model comparison in econometric modeling?

Comparing models using the same sample period and fixed T

What is the primary assumption underlying the AR(p) model in time series modeling?

The series exhibits no autocorrelation

What is the implication of positive autocorrelation in a time series?

The series wanders around its mean

What is the implication of residual autocorrelation in a time series model?

The model is not well-specified

What do non-stationary AR processes lead to?

Explosive behavior in the series

What are the Yule-Walker equations used to solve for in an AR model?

Autocorrelations and AR coefficients

What is the primary focus of the Box-Jenkins estimation approach mentioned in the handout?

Model identification, estimation, and diagnostic checking

What is the primary purpose of using time series data in economic analysis?

To analyze and forecast economic trends and indicators

What is the primary focus of diagnostic checks in econometric modeling?

To detect model misspecification and violations of assumptions

What is the connection between AR(1) and MA(∞) models through mathematical presentation?

AR(1) and MA(∞) are equivalent representations of the same process

What is the definition of an ARMA(p,q) model?

A model that combines AR(p) and MA(q) models

What is the primary focus of an ARIMA(p,d,q) process/model?

Handling non-stationary time series data

What is the Box-Jenkins procedure for estimating ARMA models primarily focused on?

Finding a parsimonious model that reflects the DGP and produces white noise error terms

What is the difficulty in distinguishing between similar ARMA models primarily attributed to?

The influence of lag length on data frequency

What is the primary purpose of the ARCH-test in time series analysis?

To test for the presence of heteroscedasticity in the error terms

What is the implication of heteroscedasticity on the estimated parameters in time series modeling?

The estimated parameters standard errors become less reliable

What is the purpose of the Newey-West HAC estimator in econometric modeling?

To compute parameter standard errors reliably in the presence of autocorrelation and heteroscedasticity

What is the primary focus of the Chow Breakpoint test in time series analysis?

To investigate whether the model parameters differ between subsamples

What is the implication of rejecting the null hypothesis in the ARCH-test?

Evidence of heteroscedasticity in the error terms

What does the Newey-West HAC estimator provide in the presence of autocorrelation and heteroscedasticity?

Reliable parameter standard errors

What is the implication of clustered or autocorrelated volatility in a time series?

The time series exhibits heteroscedasticity

What is the primary purpose of the Q-test in time series analysis?

To test for the presence of autocorrelation in the squared error terms

What is the primary implication of rejecting the hypothesis of no autocorrelation in squared residuals?

Evidence of autocorrelation in the squared error terms

What is the primary condition for the baseline IGARCH model to have a stationary variance?

$\alpha_1 + \beta_1 = 1$

What is the distinguishing feature of the EGARCH model proposed by Nelson (1991)?

It includes both positive and negative error terms in the conditional variance equation

What is the additional feature of the TARCH (GJR-GARCH) model proposed by Zakoian (1994) and Glosten et al. (1993)?

It allows for different coefficients on negative and positive error terms

What is the primary feature of the APARCH model proposed by Ding, Engle & Granger (1993)?

It allows for asymmetric effects of positive and negative shocks

What is the primary difference between ARCH and GARCH models?

ARCH models only capture autoregressive conditional heteroscedasticity, while GARCH models capture both autoregressive and generalized conditional heteroscedasticity.

What is the key contribution of Robert Engle, the winner of the 2003 Nobel Memorial Prize in Economic Sciences, to volatility modeling?

He showed that it is possible to estimate both the expected value and conditional variance of a time series simultaneously.

What is the primary focus of the ARCH-test in time series analysis?

To test for the presence of autoregressive conditional heteroscedasticity in the time series.

What is the significance of clustered volatility in financial market research?

It indicates that large shocks are more likely to be followed by large shocks, which has implications for risk management and forecasting.

What is the main reason for the substantial interest in volatility modeling in financial market research since the 1980s?

The need to accurately capture the dynamics of risk and uncertainty in financial markets.

What did Robert Engle's work demonstrate regarding the estimation of time series?

It is possible to estimate both the expected value and conditional variance of a time series simultaneously.

What is the key concept underlying GARCH models in volatility modeling?

The conditional variance of a time series is modeled as a function of past error terms and past conditional variances.

What is the primary focus of Eviews UG II, Chapter 25 mentioned in the handout?

To provide a comprehensive understanding of time series modeling and forecasting using Eviews software.

What is the preferred ARMA model for returns based on AIC & SC?

ARMA(2,1)

What type of residuals should not exhibit autocorrelation in a well-specified GARCH model?

Standardized residuals

What type of estimation is required for correct standard errors of coefficients when dealing with non-normally distributed residuals?

QML estimation

What type of model successfully captures conditional volatility with homoskedastic residuals?

GARCH(1,1)

What type of graph is used for the ARIMA(1,1,0)-GARCH(1,1) model for OMX Helsinki Small Cap index?

Conditional standard deviation (ℎ𝑡) graph

What is the purpose of using dummy variables in testing effects on conditional volatility?

To test effects on conditional volatility

What does the GARCH-in-Mean model link to conditional variance?

Risk premium

What type of test indicates heteroscedasticity, prompting continuation with GARCH modeling?

ARCH test

What type of residuals are required for correct standard errors of coefficients in econometric modeling?

Non-normally distributed residuals

What type of model is preferred over ARMA(2,3) based on AIC & SC?

AR(1)-GARCH(1,1)

What type of residuals should not exhibit autocorrelation in a well-specified model?

Standardized residuals

What type of model successfully captures conditional volatility with homoskedastic residuals?

GARCH(1,1)

Study Notes

Macroeconometrics and Time Series Data

  • Macroeconometrics focuses on forecasting models and their limitations, such as the difficulty of including quarterly GDP changes in daily stock return forecasts.
  • Macroeconomists use extensive time series data on economic fundamentals like GDP, consumption, household indebtedness, inflation, and interest rates, which can be observed in real or nominal terms and seasonally adjusted or not.
  • Nominal GDP for Finland has been observed quarterly from 1975Q1 to 2023Q2, with 194 observations.
  • Finland's private sector credit to GDP ratio has been observed quarterly from 1970Q1 to 2023Q1, with 213 observations.
  • The inflation rate in Finland, based on changes in the consumer price index, has been observed from 2000Q1 to 2020Q2, with notable seasonal variation.
  • Real vs. nominal housing price indices for the Helsinki Metro Area have been observed in natural log form from 1975Q1 to 2020Q2.
  • The S&P 500 Total Return Index has been observed daily from 11.9.1989 to 11.9.2020, with 7817 daily observations and a natural log transformed index.
  • Neste Share Return in Helsinki Stock Exchange has been observed for daily returns from 18.4.2005 to 11.9.2020.
  • Time series data is used for forecasting, estimating dynamic causal effects, modeling risks, and testing economic theories, with applications in various fields including environmental modeling and computer science.
  • Time series data raises technical issues such as time lags, correlation over time, calculation of standard errors for serially correlated errors, and data stationarity.
  • Natural logs are used in time series analysis for various reasons, including closer approximation to normal distribution and better reflecting long-term mean returns.
  • Stationarity is a key assumption in time series analysis, with the concept of covariance stationarity or weak stationarity being commonly referred to in the literature.

Time Series Analysis and Stationarity

  • Time series analysis involves verifying whether the time series is stationary or non-stationary
  • Conventional estimation techniques and testing procedures do not generally apply for non-stationary time series
  • Stationary time series are called I(0), while difference stationary time series are called I(1)
  • Some variables need to be differenced twice in order to get stationary series, and are called I(2)
  • Trend stationary time series becomes stationary after removing a deterministic trend from it
  • A trend is a persistent, long-term movement or tendency in the data
  • Deterministic trend is a nonrandom function of time, while a stochastic trend is random and varies over time
  • A random walk is a stochastic trend with serially uncorrelated disturbances
  • If a time series has a random walk trend, then ΔYt is stationary and regression analysis should be undertaken using ΔYt instead of Yt
  • Stock return index value can be a random walk with drift, a stochastic trend, which is difference stationary (I(1))
  • Autocorrelation or serial correlation is the correlation of a series with its own lagged values
  • Sample autocorrelations are estimates of the population autocorrelations, and are computed over observations in the time series.

Econometric Model Selection and Diagnostic Checks

  • In the identification stage, several alternative models are suggested for estimation.
  • Maximum Likelihood estimation is commonly applied in the estimation stage.
  • Overdifferencing is not recommended as valuable information is lost.
  • The selection of the best model considers the parsimony, statistical significance, model fit, information criteria, out-of-sample forecast accuracy, and diagnostic checks.
  • Parsimonious models typically produce more accurate forecasts and should generally have statistically significant parameters.
  • Information criteria such as Akaike and Schwartz are commonly used for model selection.
  • Comparing models using the same sample period and fixed T in the estimations is essential.
  • Diagnostic checks aim to detect whether the model reflects the DGP well and if the error term is white noise.
  • Visual inspection and formal tests are used to check for outlier observations, structural changes, autocorrelation, and residual normality.
  • Testing for autocorrelation is particularly important as evidence of residual autocorrelation implies that the model does not cater for some systematic process in the time series.
  • Residual normality is often assumed, and formal testing such as the Jarque-Bera test is applied.
  • Non-normality of the error term can lead to unreliable confidence intervals and p-values, and can be controlled using point dummy variables or the Student's t-distribution.

Econometric Model Selection and Diagnostic Checks

  • In the identification stage, several alternative models are suggested for estimation.
  • Maximum Likelihood estimation is commonly applied in the estimation stage.
  • Overdifferencing is not recommended as valuable information is lost.
  • The selection of the best model considers the parsimony, statistical significance, model fit, information criteria, out-of-sample forecast accuracy, and diagnostic checks.
  • Parsimonious models typically produce more accurate forecasts and should generally have statistically significant parameters.
  • Information criteria such as Akaike and Schwartz are commonly used for model selection.
  • Comparing models using the same sample period and fixed T in the estimations is essential.
  • Diagnostic checks aim to detect whether the model reflects the DGP well and if the error term is white noise.
  • Visual inspection and formal tests are used to check for outlier observations, structural changes, autocorrelation, and residual normality.
  • Testing for autocorrelation is particularly important as evidence of residual autocorrelation implies that the model does not cater for some systematic process in the time series.
  • Residual normality is often assumed, and formal testing such as the Jarque-Bera test is applied.
  • Non-normality of the error term can lead to unreliable confidence intervals and p-values, and can be controlled using point dummy variables or the Student's t-distribution.

Time Series Analysis and ARMA Models

  • MA(1) process examples with different parameters and autocorrelation values
  • Example of MA(2) process with parameters and autocorrelation value based on true data
  • Mean and variance equations for MA(1) and MA(q) processes
  • Explanation of autocorrelation in MA(q) process and its drop to zero after q lags
  • Theoretical ACF and PACF for MA(1) process and its application to real-life data
  • Connection between AR(1) and MA(∞) models through mathematical presentation
  • Explanation of ARMA(p,q) model as a combination of AR(p) and MA(q) models
  • Difficulty in distinguishing between similar ARMA models and the influence of lag length on data frequency
  • Definition and components of ARIMA(p,d,q) process/model
  • Difficulty in identifying correct p and q for ARMA process and starting estimation with ARMA(1,1) model
  • Inclusion of other explanatory variables in ARIMA model and discussion on seasonal variation
  • Box-Jenkins procedure for estimating ARMA models, including identification and estimation steps, and the goal of finding a parsimonious model that reflects the DGP and produces white noise error terms

Estimating GARCH Models and GARCH-in-Mean Models

  • ARMA(2,|3|) for returns, based on Box-Jenkins procedure, but not normally distributed error term with fat tails, potentially due to heteroscedasticity
  • Addition of Covid19 dummy has minimal impact on model
  • Both Q-test and ARCH test indicate heteroscedasticity, prompting continuation with GARCH modeling
  • Preference for AR(1) over ARMA(2,|3|) based on AIC & SC, resulting in AR(1)-GARCH(1,1) model
  • Standardized residuals Zt should not exhibit autocorrelation, ensuring well-specified model
  • Non-normally distributed Zt requires QML estimation for correct std. errors of coefficients
  • GARCH(1,1) model successfully captures conditional volatility with homoskedastic residuals
  • Non-normal residuals after QML estimation, but coefficients remain statistically highly significant
  • Conditional standard deviation (ℎ𝑡) graphed for ARIMA (1,1,0)-GARCH(1,1) model for OMX Helsinki Small Cap index
  • Use of dummy variables to test effects on conditional volatility (e.g., Covid19) considered
  • Unconditional vs. conditional mean and variance concepts explained in the context of GARCH models
  • Introduction and explanation of GARCH-in-Mean model, which links risk premium to conditional variance

Test your knowledge of macroeconometrics, time series data, and stationarity in this quiz. Explore forecasting models, time series data on economic fundamentals, and technical issues in time series analysis. Gain insights into the concepts of stationarity, trend stationary time series, and autocorrelation.

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